Relation between classes of language in chomsky hierarchy :
The following theorem states the relationships between classes of languages in chomsky hierarchy, namely- type-0 or r.e. languages, type-1 or CSLs, type-2 or CFLs and type-3 or regular languages.
Theorem: For each i=0, 1, 2 the class of type-i language properly includes the class of type-(i+1) languages.
- The containments are clear from the fact that type-(i+1) grammar is a restricted form of type-i grammar for each i=0, 1, 2.
- we have already seen that
is a type-2 language (or context-free language) that is not regular. Thus the class of type-2 language properly includes the class of type-3 language (or regular languages). - we have also shown that the language
is a type-1 or context-sensitive language but not a type-2 language (or CFL). Hence, the class of type-3 languagee properly includes the class of type-2 languages. - we will use the following two theorems to show that the class type-0 languages properly includes the class of type-1 langauges. This will complete the proof of the above theorems.